This paper develops a unified framework for the economic analysis of envy by capturing its two main forces, destructive and competitive. The role of envy in society is determined in equilibrium by the level of fundamental inequality and the endogenous tolerance threshold shaped by the quality of institutions and the strength of positional concerns.
The qualitatively different equilibria that arise in this framework are broadly consistent with the available evidence on the implications of envy for economic incentives. The
“keeping up with the Joneses” equilibrium roughly corresponds to modern economies, in which emulation is the main driver of consumer demand. The “fear equilibrium” resembles the role of envy in developing countries, where the fear of envious retaliation prevents productive investment and retards progress. The different nature of these equilibria yields contrasting comparative statics with respect to the strength of positional concerns. In the KUJ equilibrium, envy enhances production by intensifying emulation, while in the fear equilibrium it reduces output by aggravating the fear constraint.
The model sheds new light on the interplay between institutions, welfare, and economic performance. First, while better institutions can move the society from the low-output fear equilibrium to the high-output KUJ equilibrium, such change need not be welfare enhancing if it triggers the “rat race” competition. Second, transfers can be used to avoid envy-motivated destruction in equilibrium, which is in line with the evidence on the role of redistributive mechanisms in developing societies.
A dynamic extension of the model explores the evolution of the economy through differ-ent envy regimes. Starting at any initial conditions, the economy converges to the long-run KUJ steady state, driven by the decline of inequality along the transition path. This tran-sition can be delayed or accelerated by factors that affect the strength of social comparisons and the relative attractiveness of productive and destructive effort.
This paper lays a foundation for understanding the changing role of envy in the process of development. This line of research is pursued in Gershman (2011b) by incorporating the basic model in a simple endogenous growth framework, in which resources are lim-ited and productivity is driven by learning-by-doing and knowledge spillovers. As rising productivity expands the production possibilities frontier for everyone, the society experi-ences an endogenous transition from the fear equilibrium to the KUJ equilibrium causing a qualitative change in the relationship between envy and economic performance.
Appendix
τ θ <1 andY2> Y1, and Lemma 1 gives the unique second-stage equilibrium.Proof of Lemma 2. Agent 2 is solving (10) subject to (8) and (9). Consider first the caseτ θY26Y1, in
It is strictly concave inY2, and the first-order conditions yield the following optimum:
Y2=
Again,U2is strictly concave inY2, and the interior optimum is uniquely defined by the first-order condition rY1Y2
To rewrite (A2) in terms of consumption note that if destruction takes place
Substituting this into (A2) yields (12). Next, applying the implicit function theorem to (12) gives dC2
sinceC2> θC1 and 0< dC2/dC1< C2/C1. Hence,C2d(C1) is strictly increasing and concave.
It is strictly concave inY1, and the first-order conditions yield the following optimum:
Y1=
Assumptionσ >1 is sufficient forU1to be strictly concave inY1. In particular, it can be shown by simple differentiation that the sign ofd2U1/dY12coincides with the sign ofh(x)≡ −(1+σ)x2+ξ(2σ+1/2)x−σξ2, wherex≡p
θY2/τ Y1 andξ≡(τ+ 1)/τ. Then, it is easy to show that hmax∝1−8σ <0 and so,U1 is concave underσ >1.
Note that U1 is differentiable at point Y1 = τ θY2 with dU1(τ θY2)/dY1 = ((τ −1)θY2)−σ−1/K1. Hence, there are only two cases. IfY26Cb2, the global optimum is given by the interior optimum in the KUJ region. IfY2>Cb2, there global optimum is in the destructive region and it is given by the first-order condition
Finally, using (A3) rewrite equation (A6) in terms of consumption levels to get (15). Applying the implicit function theorem to (15) gives
Proof of Proposition 1. Private outputs in the KUJ equilibrium are given by (16). For this to be an equilibrium two conditions must hold: Y1KUJ>Ce1 andY2KUJ6Cb2. This yieldsk>k, where ˜˜ kis defined in Proposition 1. In the fear equilibrium private outputs are given by (17). ForY2Fto be the best response of Agent 2 it must be the case thatCb16Y1F<Ce1, which yields ˆk6k <k˜ with ˆkdefined in Proposition 1. The borderline fear equilibrium happens when τ θCb2 =Cb1, that is, whenk = ˆk. If τ θCb2 <Cb1, that is, k <k, the only possibility is the destructive equilibrium. To prove its existence note that: 1) fromˆ Lemma 2,C2d(C1) is strictly increasing and concave which implies thatC2d(C1d(Cb2))>Cb2; 2) from Lemma
3,C1d(C2) is strictly increasing and convex which implies that the inverse ofC1d at pointCb1is greater than C2d(Cb1) =Cb1/τ θ. Hence, the intermediate value theorem guarantees existence of a destructive equilibrium (C1D, C2D) such that C1D<Cb1 andC2D>Cb2. Finally, direct comparison of (A4) and (A7) shows that the slope of the inverse ofC1d is always steeper than that ofC2d, which ensures single crossing.
Proof of Proposition 2. Results for the KUJE and the FE follow directly from (16) and (17). For the DE, consider the system defining the equilibrium:
(f1≡C1−θC2−ψ[C1/(C1+θC2)]1/σ= 0;
f2≡C2−θC1−φ[(C1+θC2)/C2]1/σ= 0.
Using the equilibrium conditions, it can be shown that [DCf]−1= 1
It follows that the sign of∂C1/∂λcoincides with the sign of
Plugging this expression in the previous equation and making transformations, we get σ(θ+x) +x(1−θx)
The first term is always greater than 1. The second term is increasing in x and at x= 1 simplifies to τ(1 +θ2)(1 +θ)2/[2(1 +τ)2]. The latter expression is maximized at τ = 1 in which case it is equal to (1 +θ2)(1 +θ)2/461∀ θ∈(0,1). Hence, the second term is always less than 1 and∂C1/∂λ >0.
Although the sign of∂C2/∂λis ambiguous, the sign of∂C/∂λcan be determined, as it coincides with the sign of
As above, the first term of this expression is always greater than 1, while the second is always less than 1.
Hence,∂C/∂λ >0.
For part (b), note thatDθC=−[DCf]−1·Dθf and
It follows that the sign of∂C1/∂θ is determined by the sign of C2+θC1− 1 main text, the sign of∂C2/∂θ is generally ambiguous.
For part (c), note thatDτC=−[DCf]−1·Dτf and
with both elements positive, sinceψτ <0 andφτ <0. It follows that the elements ofDτC are negative meaning thatC1and C2 are decreasing inτ.
Proof of Proposition 3. As follows from (17), the utilities in the FE are given by UF1= In the KUJE, as follows from (16), they are
UKUJi = LetL(k) andR(k) denote the left-hand side and the right-hand side of (A10), respectively. Note that: 1) R(k) is strictly increasing and concave, 2) L(˜k) =R(˜k), and 3)L′(˜k)> R′(˜k). Hence, eitherL(k)< R(k)
such thatg(z)>0 iffz >z. This implies that¯ ∃! ¯θ∈(0,1), such thatL(k)< R(k) (and thus,UKUJ2 < UF2)
∀k∈(ˆk,k) iff˜ θ >θ. If¯ θ <θ,¯ ∃! ¯k∈(ˆk,k), such that˜ UKUJ2 < UF2iffk >k, which completes the proof.¯ Proof of Proposition 4. It follows immediately from (A8) thatdUF1/dk >0 sinceK1=λK =kK/(1+k).
As for Agent 2,
Differentiating with respect tokand re-arranging terms we obtain thatdUF2/dk >0 iffh(k)≡σk2+k−δ <
0, whereδis defined in Proposition 4. Next,h(ˆk)>0 iff
Finally (A9) can be re-written in terms ofkas UKUJ1 =−
Proof of Lemma 4. Using (8) and (9), rewrite (22) as v (1 +θ2) The objective function is strictly concave inT. The first-order condition for interior solution, T∗, is
1 +θ2
where γ2 is defined in Lemma 5. It is strictly concave in Y2 and the unique optimum is given by Y2 = (γ2K2)1/σ −Y1, which is interior iff Y1 < Y˘1, where ˘Y1 is defined in Lemma 5, and ˘Y1 < Ye1, as direct comparison shows. This yields (23).
Proof of Lemma 6. In the KUJ case,τ θY2 6Y1, as established in the proof of Lemma 3, the optimal action of Agent 1 is (A5). Now consider the case in which Agent 2 makes a full transfer under a credible threat of destruction,τ θY2> Y1. The utility of Agent 1 is
U1= γ1(Y1+Y2)1−σ 1−σ − Y1
K1
, whereγ1 is defined in Lemma 5. This yields the following solution:
Y1=
where ˘Y2 is defined in Lemma 5. First, note that if the optimum is not interior in the KUJ region, that is, ifY2 >Cb2, the global optimum is determined by (A11) since the left derivative of U1 at point Y1=τ θY2 is less than the right derivative. In particular, dU1(τ θY2−)/dY1 =γ1(1 +τ θ)−σY2−σ−1/K1, while dU1(τ θY2+
)/dY1 = (τ θY2−θY2)−σ−1/K1, and, after re-arrangement, the former is smaller than the latter iffγ1<[(1 +τ θ)/θ(τ−1)]σ which always holds since θ(τ−1)<1 +τ θ.
Consider for concreteness only the case ˘Y2<Cb2 (the other case can be analyzed in exactly the same way) which holds if
γ11/σ < 1
θ(τ−1). (A12)
Then, as follows from (A11), Y2 >Cb2 implies that the best response will be b3. If, however, Y2 <Cb2, the optimum in the KUJ region is interior and needs to be compared to the best response in the transfer region. In the former case
The left-hand side of this expression, L(Y2) is strictly decreasing in Y2, which implies that the above inequality always holds in the region ˘Y26Y2<Cb2if it holds at pointY2= ˘Y2. The latter yields condition (A13). Assume that it holds along with (A12), in which caseY1= 0 is the best response for Y2>Y˘2.
Next, consider the region ˘Y2/(1 +τ θ) 6Y2 <Y˘2, in which b2 is the best response in the case with transfer and the corresponding utility is
UFT1 = σ
Since (A13) holds and KUJ optimum is interior (that is,b1 cannot be a best response) it must be the case that in this region there exists a value ofY2 such that UFT1 =UKUJ1 . Direct computation shows that this value isYe2, as defined in Lemma 5, where ˘Y2/(1 +τ θ)<Ye2<Y˘2. So, the global optimum is the KUJ best response, ifY26Ye2, andb2, otherwise. Putting everything together yields the statement of Lemma 5.
Proof of Proposition 5. For (16) to be an equilibrium, as follows from Lemma 5 and Lemma 6, the following conditions must hold: Y1KUJ > Ye1 and Y2KUJ 6 Ye2. The former yields the condition k >
(γ2/γ1)σ/(σ−1), while the latter leads tok>ω. As follows from the lemmas below, ω is well-defined and exceeds (γ2/γ1)σ/(σ−1), that is, a KUJ equilibrium exists iffk>ω.
Lemma A. Under the full transfer condition, τ θ 6(1−θ)/(1 +θ), the parameter ω is well-defined, that is,σ(1−γ11/σ)(1−θ)> θ(σ−1). The latter can be re-arranged as (σ−θ)/(1−θ)< σγ11/σ. Define θ/(1−θ). The right-hand side, R(a), is strictly increasing and convex in a, with R(0) = 1 = L(0), R(1) = (1 +τ θ)/(θ(τ−1))> L(1), andR′(0) = ln[(1 +τ θ)/θ(τ−1)]. Note thatR′(0) is strictly decreasing in τ and, hence, under the full-transfer assumptionτ <[1−θ]/[θ(1 +θ)], infR′(0) = ln[2/(1−2θ−θ2)].
The latter is well-defined sinceθ <√
2−1, as follows from the full-transfer assumption together withτ >1.
Finally, ln[2/(1−2θ−θ2)] monotonically increases inθon the segment [0,√ afrom Lemma A, this inequality can be re-written as
1 +τ θ θ(τ−1)
a
<1 + a(1 +θ) θ(τ−1).
The left-hand side is a strictly increasing convex function of a, while the right-hand side is linear in a.
Moreover, the two parts coincide at the endpoint of the [0,1] segment. It follows that the above inequality holds for alla∈(0,1).
For (25) to be an equilibrium the following conditions must hold: Y1FT 6 Y˘1 and Y2FT > Y˘2. The former is always true, while the latter yields the condition k 6 γ2/γ1. It follows that FT and KUJ equilibria coexist iff ω 6 k 6 γ2/γ1, where ω < γ2/γ1 is the existence assumption. If k = γ2/γ1, the best-response functions partially overlap yielding a continuum of equilibria.
Finally, it can be shown that the fear equilibrium without transfers (as in the basic model) does not exist. For (17) to be an equilibrium, conditionY2F=Cb26Ye2must be satisfied, but restriction (A12) rules this out sinceYe2<Y˘2<Cb2. Putting everything together yields the statement of Proposition 5.
Proof of Proposition 6. It follows from (16) and (25) that C1KUJ
C2KUJ = k1/σ+θ
1 +θk1/σ, C1FT C2FT =τ θ.
Then, inequality is higher in FT equilibrium iffk > [(1−τ θ2)/θ(τ−1)]−σ, and it is sufficient to show
The functionq(a) is strictly increasing since q′(a) =a which completes the proof of part 1.
It follows from (16) and (25) thatYKUJ> Y2FT iff (k1/σ+ 1)/(1−θ)> γ1/σ2 . Sincek > ω, k1/σ+ 1
1−θ > ω1/σ+ 1
1−θ = σγ11/σ−1
σ(γ11/σ−1)−θ(σγ11/σ−1).
As can be easily shown, the right-hand side of this expression exceedsγ21/σ iff 1−θ < h(a), where h(a)≡ 1
As follows from (2), (5), and Proposition 5, in the FT equilibrium the utilities of agents are given by:
UFT1 = γ1(γ2K2)1−σσ
In the KUJ equilibrium they are given by (A9). Direct comparison shows that UKUJ1 > UFT1 iff (1− θ2)γ1γ2(1−σ)/σk >(σ−θ2)k1/σ+θ(σ−1), which is the condition in Proposition 6. Similarly, UKUJ2 > UFT2
Since the left-hand side is strictly increasing in k, it is sufficient to prove that the above inequality holds at k =γ2/γ1 to establish that it holds ∀k ∈ (ω, γ2/γ1). After re-arranging the terms, at k =γ2/γ1 the
Denote the right-hand side of this inequality asR(a). Then,R′(a)>0 iff q(a)≡
The functionq(a) is strictly increasing since (29). The form of g3(kt) follows from (12), (15) and (29). Next, it is straightforward to establish by differentiation thatg1(k) andg2(k) are strictly increasing and concave, given thatβ <1 andσ(1−β)>1.
Furthermore, 1)g1(1) = 1, 2) g1(˜k) =g2(˜k)>k, and 3)˜ g2(ˆk) =g3(ˆk)>ˆk. The second property holds iff τ θ > (˜k)1−β = [θ(τ −1)/(1−τ θ2)]σ(1−β) which is always trues since τ θ > θ(τ −1)/(1−τ θ2) and σ(1−β)>1. The third property holds iffτ θ >(ˆk)1−β which is always true, as follows from property 2 and the fact that ˆk <˜k. Together these properties imply thatkt+1> kt forkt>k.ˆ
Finally, if there exists a steady state ¯kin the destructive segment, it is given byg3(¯k) = ¯k. Making a substitutionκ= ¯k1−β and rearranging terms yields the equation defining the steady state:
κ2−β right-hand side of (A14), is strictly increasing and convex. Also, L(κ), the left-hand side of (A14), is strictly increasing. To show that it is concave forκ > θ, note that the sign ofL′′(κ) is determined by the sign of L(κ)˜ ≡(κ+θ)(2−β)(βκ+θ(1−β))−κ(3−2β)(βκ+θ(2−β)). Atκ=θthis expression is negative, since β <1. Moreover, it is strictly decreasing for the same reason. Hence,L′′(κ)<0 forκ > θ. This implies that there exists at most one solution to (A14). If there is no solution, the proof is finished. Assume now that ¯k exists. In this case ¯k >k. To see this, note first that ˆˆ k <(τ θ)1/(1−β). Next, κ > τ θ since L(τ θ)> R(τ θ). Hence, ¯k≡κ1/(1−β)>(τ θ)1/(1−β)>ˆk, that is,kt+1> ktfor 0< kt<k.ˆ
Proof of Lemma 9. It follows from the properties of g1(kt) that kt monotonically converges to 1 in the KUJ region. The expression for ∆Ki = 0 comes from (30) and may be rewritten as Kjt = Kit[(ρKit1−β−1/σ −1)/θ]σ, where ρ ≡ (1−θ2)/α. Then, by differentiation we get that dKjt/dKit > 0 and d2Kjt/dKit2 >0, since σ(1−β)> 1. This implies the stated properties of the ∆Ki = 0 loci. The expression for ¯K follows from solving ∆K1= ∆K2= 0 and (16) then gives ¯Y.
Proof of Lemma 10. The equations for ∆Ki = 0 come from (30). Since in the fear region K1t+1 = ατ K1t1/σ+β/(τ −1) and σ(1−β)> 1,K1t converges to ¯K1. Plugging this in ∆K2 = 0 yields ¯K2. The output levels follow from (17). To see that the long-run fear equilibrium is in the KUJ region, note that the implied steady-state level of inequality is [τ θ]1/(1−β) which exceeds ˜k = [θ(τ−1)/(1−τ θ2)]σ, since σ(1−β)>1 andθ(τ−1)/(1−τ θ2)< τ θ <1.
Equilibria in the model with symmetric destruction opportunities. As mentioned in section 3, the assumption about “predator and prey” relationship between Agent 1 and Agent 2 is without loss of generality. Combining the proofs of lemmas 2 and 3 we get that in the general case the best responses of both agents are symmetric and have the following form:
Ci∗(Cj) =
It follows from the proof of Proposition 1 and the fact thatK1< K2 that the equilibria of this game in general formulation are fully characterized in Proposition 1. Figure 14 depicts the alternative equilibria under symmetric best responses (cf. figures 3 and 4).
BR2
Figure 14: Equilibria in the symmetric envy game.
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